Theorem isorcl2 49387

Description: Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
isorcl.i	⊢ 𝐼 = (Iso‘𝐶)
isorcl.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
isorcl2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
isorcl2	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Proof of Theorem isorcl2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isorcl.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
2		isorcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2737	. . . . 5 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		isorcl.i	. . . . . 6 ⊢ 𝐼 = (Iso‘𝐶)
5	4, 1	isorcl 49386	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	2, 3, 5, 4	isofval2 49385	. . . 4 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦)))
7	6	oveqd 7385	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌))
8	1, 7	eleqtrd 2839	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌))
9		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))
10	9	elmpocl 7609	. 2 ⊢ (𝐹 ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
11	8, 10	syl 17	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  dom cdm 5632  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  Invcinv 17681  Isociso 17682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-inv 17684  df-iso 17685

This theorem is referenced by:  isoval2  49388  catcisoi  49753